The problem of detecting the existence of a changepoint in a data sequence and of identifying its position is challenging when the focus is on extreme events and the distribution of data is heavy-tailed. In this setting, we recall several existing techniques, and then we propose a novel robust semi-parametric approach to changepoint identification that does not require the likelihood function. The changepoint is estimated as the position of the maximum of a statistic inspired by classical ANOVA to contrast the tail behavior of data to the left and right of all changepoint candidates. It is shown that the estimator consistently identifies the true changepoint under mild assumptions when the sample size increases. In numerical experiments, the novel method shows reliable finite-sample behavior for various simulation settings and is very competitive in comparison to the alternative changepoint identification approaches from the literature, especially for small sample sizes. Finally, the utility of the method is highlighted by identifying interpretable changepoints in three real-data applications: very large motor insurance claim amounts for a French administrative region with age as covariate; daily Bitcoin cryptocurrency price data and daily returns of stocks of the Boeing company both with time as covariate.
While random probability measures have a long tradition in probability and statistics, little is known about their tails. The few available results are derived using subordinators, and therefore only apply to measures that can be represented as normalized subordinators, such as the Dirichlet process. Our work breaks this barrier, by exploiting the stick-breaking representation to construct a new family of transport maps that preserve the decay of tails. Drawing on recent developments on regular variation and on subordinator theory, the new family of maps allows us to establish that the right tail of a Pitman–Yor process is heavy-tailed if the centering distribution is itself heavy-tailed; the Dirichlet process is the only member of this class that fails to obey this convenient property. Asymptotic envelopes for the tails of the Pitman–Yor processes are also derived. Finally, we discuss some consequences of the main results, including aspects related to the posterior distribution.
We correct and generalise Theorem 3 in Novak (2019), and formulate an open question.
This editorial opens the special issue. While the origins of AI (Artificial Intelligence) can be traced back to the 1920s and 1930s—with foundational developments by figures such as Gödel and Turing in logic and theoretical computer science—the roots of EVT (Extreme Value Theory) reach back to roughly the same period, marked by pioneering work from Fisher and Tippett. Despite these parallel beginnings, it has taken nearly a century for the two fields to begin meaningfully intersecting. This special issue aims to highlight that the time is now ripe for deeper mutual engagement, with the potential for these synergies to generate novel questions and solutions across both domains.
Investigating the level sets L^t of Archimax copulas C ∈𝒞_am , we establish that these sets can be characterized in terms of certain convex functions f^s and non-decreasing functions g^t . Motivated by the results in Mai and Scherer (Extremes 14, 311-324 2011) and Trutschnig et al. (Extremes 19, 405-427 2016), which examine the way bivariate Extreme Value copulas distribute their mass, we extend these findings to the larger family of bivariate Archimax copulas C_am . Working with Markov kernels (conditional distributions), we analyze the mass distributions of Archimax copulas C ∈𝒞_am and show that the support of C is determined by some functions f^0 , g^L , and g^R . Additionally, we prove that the discrete component (if any) of C concentrates its mass on the graphs of the afore-mentioned functions f^s or g^t . Recognizing the close relationship between the level sets L^t of a copula C and its Kendall distribution function F_C^K , we provide an alternative proof for the representation of F_C^K for arbitrary Archimax copulas C ∈𝒞_am and derive simple expressions for the level set masses μ _C(L^t) . Building upon the fact that Archimax copulas C ∈𝒞_am can be represented via two univariate probability measures γ and ϑ — so-called Williamson and Pickands dependence measures — we show that absolute continuity, discreteness, and singularity properties of these measures γ and ϑ carry over to the corresponding Archimax copula C_γ , ϑ . Finally, we derive conditions on γ and ϑ such that the support of the absolutely continuous, discrete, or singular component of C_γ , ϑ coincides with the support of C_γ , ϑ .
This paper devises a novel neural-inspired approach for simulating multivariate extremes. Specifically, we propose a GAN-based generative model for sampling multivariate data exceeding large thresholds, giving rise to what we refer to as the ExceedGAN algorithm. Our approach is based on approximating marginal log-quantile functions using feedforward neural networks with eLU activation functions specifically introduced for bias correction. An error bound is provided on the margins, assuming a J th order condition from extreme value theory. The numerical experiments illustrate that ExceedGAN outperforms competitors, both on synthetic and real-world data sets.
The dimension of random simplicial complexes (defined as the maximal dimension among all faces) is a natural extreme value associated with the complex, and is closely related to other functionals defined by a maximum, such as the clique number of geometric graphs or scan statistics. We extend existing results in the binomial point process case to the Poisson setting in sparse graphs, give new ones about expectations and large deviation principles in all regimes, as well as give a first precise distribution result in the dense case.
Non-stationary extremal dependence, whereby the relationship between the extremes of multiple variables evolves over time, is commonly observed in many environmental and financial data sets. However, most multivariate extreme value models are only suited to stationary data. A recent approach to multivariate extreme value modelling uses a geometric framework, whereby extremal dependence features are inferred through the limiting shapes of scaled sample clouds. This framework can capture a wide range of dependence structures, and a variety of inference procedures have been proposed in the stationary setting. In this work, we first extend the geometric framework to the non-stationary setting and outline assumptions to ensure the necessary convergence conditions hold. We then introduce a flexible, semi-parametric modelling framework for obtaining estimates of limit sets in the non-stationary setting. Through rigorous simulation studies, we demonstrate that our proposed framework can capture a wide range of dependence forms and is robust to different model formulations. We illustrate the proposed methods on financial returns data and present several practical uses.
It is increasingly the case with modern time series that many data sets of practical interest contain abrupt changes in structure. These changes may occur in complex characteristics such as the extremal dependence structure, and identifying such structural breaks remains a challenging problem. Many existing change point detection algorithms focus on changes in dependence across the entire distribution, rather than the tails, and approaches that are tailored to extremes typically make strict parametric assumptions or they are only applicable to bivariate data. We propose a nonparametric MOving sum-based approach for detecting multiple changes in the Pairwise Extremal Dependence (MOPED) of multivariate regularly varying data. To avoid the classical problem of threshold selection in the study of multivariate extremes, we further propose a multiscale, multi-threshold variant of MOPED that pools change point estimates across choices of the threshold and the bandwidth used in local estimation. Good performance of MOPED is illustrated in a simulation study, and we showcase its ability to identify subtle changes in tail dependence class in the absence of correlation changes. We further demonstrate the usefulness of MOPED by identifying changes in the extremal connectivity of electroencephalogram (EEG) signals of seizure-prone neonates.
A common object to describe the extremal dependence of a d-variate random vector X is the stable tail dependence function L. Various parametric models have emerged, with a popular subclass consisting of those stable tail dependence functions that arise for linear and max-linear factor models with heavy tailed factors. The stable tail dependence function is then parameterized by a d × K matrix A, where K is the number of factors and where A can be interpreted as a factor loading matrix. We study estimation of L under an additional assumption on A called the ‘pure variable assumption’. Both K ∈{1, … , d} and A ∈ [0, ∞ )^d × K are treated as unknown, which constitutes an unconventional parameter space that does not fit into common estimation frameworks. We suggest two algorithms that allow for estimation of K and A, and provide finite sample guarantees for both algorithms. Remarkably, the guarantees allow for the case where the dimension d is larger than the sample size n. The results are illustrated with numerical experiments and two case studies.
We extend the Lévy Langevin Monte Carlo method studied by Oechsler in 2024 to the setting of a target distribution with heavy tails: Choosing a target distribution from the class of subexponential distributions we prove convergence of a solution of a stochastic differential equation to this target. Hereby, the stochastic differential equation is driven by a compound Poisson process - unlike in the case of a classical Langevin diffusion. The method allows one to sample from non-smooth targets and distributions with separated modes with exponential convergence to the invariant distribution, which in general cannot be guaranteed by the classical Langevin diffusion in presence of heavy tails. The method is promising due to the possibility of a simple implementation because of the compound Poisson noise term.
This paper addresses the growing concern of cascading extreme events, such as an extreme earthquake followed by a tsunami, by presenting a novel method for risk assessment focused on these domino effects. The proposed approach develops an extreme value theory framework within a Kolmogorov-Arnold network (KAN) to estimate the probability of one extreme event triggering another, conditionally on a feature vector. An extra layer is added to the KAN architecture to ensure that the parameter of interest lies within the unit interval, and we refer to the resulting neural model as KANE (KAN with Natural Enforcement). The proposed method is backed by exhaustive numerical studies and further illustrated with real-world applications to seismology and climatology.
Conformal prediction is a popular method to construct prediction intervals with marginal coverage guarantees from black-box machine learning models. In applications with potentially high-impact events, such as flooding or financial crises, regulators often require very high confidence for such intervals. However, if the desired level of confidence is too large relative to the amount of data used for calibration, then classical conformal methods provide infinitely wide, thus, uninformative prediction intervals. In this paper, we propose a new method to overcome this limitation. We bridge extreme value statistics and conformal prediction to provide reliable and informative prediction intervals with high-confidence coverage, which can be constructed using any black-box extreme quantile regression method. A weighted version of our approach can account for nonstationary data. The advantages of our extreme conformal prediction method are illustrated in a simulation study and in an application to flood risk forecasting.
Economically responsible mitigation of multivariate extreme risks-such as extreme rainfall over large areas, large simultaneous variations in many stock prices, or widespread breakdowns in transportation systems-requires assessing the resilience of the systems under plausible stress scenarios. This paper uses Extreme Value Theory (EVT) to develop a new approach to simulating such multivariate extreme events. Specifically, we assume that after transformation to a standard scale the distribution of the random phenomenon of interest is multivariate regular varying and use this to provide a sampling procedure for extremes on the original scale. Our procedure combines a Wasserstein-Aitchison Generative Adversarial Network (WA-GAN) to simulate the tail dependence structure on the standard scale with joint modeling of the univariate marginal tails on the original scale. The WA-GAN procedure relies on the angular measure-encoding the distribution on the unit simplex of the angles of extreme observations-after transformation to Aitchison coordinates, which allows the Wasserstein-GAN algorithm to be run in a linear space. Our method is applied both to simulated data under various tail dependence scenarios and to a financial data set from the Kenneth French Data Library. The proposed algorithm demonstrates strong performance compared to existing alternatives in the literature, both in capturing tail dependence structures and in generating accurate new extreme observations.
Semi-parametric quantile regression (SPQR) is a flexible approach to density regression that learns a spline-based representation of conditional density functions using neural networks. As it makes no parametric assumptions about the underlying density, SPQR performs well for in-sample testing and interpolation. However, it can perform poorly when modelling heavy-tailed data or when asked to extrapolate beyond the range of observations, as it fails to satisfy any of the asymptotic guarantees provided by extreme value theory (EVT). To build semi-parametric density regression models that can be used for reliable tail extrapolation, we create the blended generalised Pareto (GP) distribution, which i) provides a model for the entire range of data and, via a smooth and continuous transition, ii) benefits from exact GP upper-tails without the need for intermediate threshold selection. We combine SPQR with our blended GP to create semi-parametric quantile regression for extremes (SPQRx), which provides a flexible semi-parametric approach to density regression that is compliant with traditional EVT. We handle interpretability of SPQRx through the use of model-agnostic variable importance scores, which provide the relative importance of a covariate for separately determining the bulk and tail of the conditional density. The efficacy of SPQRx is illustrated on simulated data, and an application to U.S. wildfire burnt areas from 1990-2020.
The masses of data now available have opened up the prospect of discovering weak signals using machine-learning algorithms, with a view to predictive or interpretation tasks. As this survey of recent results attempts to show, bringing multivariate extreme value theory and statistical learning theory together in a common, nonparametric and nonasymptotic framework makes it possible to design and analyze new methods for exploiting the scarce information located in distribution tails in these purposes. This article reviews recently proved theoretical tools for establishing guarantees for supervised or unsupervised algorithms learning from a fraction of extreme data. These are mainly exponential maximal deviation inequalities tailored to low-probability regions and concentration results for stochastic processes empirically describing the behavior of multivariate extreme observations, their dependence structure in particular. Under appropriate assumptions of regular variation, several illustrative applications in multivariate settings are then examined: classification, regression, anomaly detection, model selection via cross-validation. For these, generalization results are established inspired by the classical bounds in statistical learning theory. In the same spirit, it is also shown how to adapt the popular high-dimensional Lasso technique in the context of extreme values for the covariates with generalization guarantees.
This note is an addendum to the work initiated by Eberlein, Kabanov, and Schmidt and developed further by Kabanov and Promyslov on the asymptotics of the ruin probabilities in the Sparre Andersen model with investments in a risky asset. Using more advanced methods of the implicit renewal theory, we provide complements to some results of the mentioned works.
We study the consistency and weak convergence of the conditional tail function and conditional Hill estimators under broad dependence assumptions for a heavy-tailed response sequence and a covariate sequence. Consistency is established under α-mixing, while asymptotic normality follows from β-mixing and second-order conditions. A key aspect of our approach is its versatile functional formulation in terms of the conditional tail process. Simulations demonstrate its performance across dependence scenarios. We apply our method to extreme event modelling in the oil industry, revealing distinct tail behaviours under varying conditioning values.
We employ the novel theory of heterogeneous extreme value statistics to accurately estimate the ultimate world records for the 100-m running race, for men and for women. For this aim we collected data from 1991 through 2023 from thousands of top athletes, using multiple fast times per athlete. We consider the left endpoint of the probability distribution of the running times of a top athlete and define the ultimate world record as the minimum, over all top athletes, of all these endpoints. For men we estimate the ultimate world record to be 9.56 seconds. More prudently, employing this heterogeneous extreme value theory we construct an accurate asymptotic 95% lower confidence bound on the ultimate world record of 9.49 seconds, still quite close to the present world record of 9.58. For the women's 100-meter dash our point estimate of the ultimate world record is 10.34 seconds, somewhat lower than the world record of 10.49. The more prudent 95% lower confidence bound on the women's ultimate world record is 10.20.
In topological data analysis, the notions of persistent homology, birthtime, lifetime, and deathtime are used to assign and capture relevant cycles (i.e., topological features) of a point cloud, such as loops and cavities. In particular, cycles with a large lifetime are of special interest. In this paper, we study such large-lifetime cycles when the point cloud is modeled as a Poisson point process. First, we consider the case with no bound on the deathtime, where we establish Poisson convergence of the centers of large-lifetime cycles on the 2-dimensional flat torus. Afterwards, by imposing a bound on the deathtime, we enter a sparse connectivity regime, and we prove joint Poisson convergence of the centers, lifetimes, and deathtimes in dimensions d ⩾ 2 under suitable model conditions.
